Asymptotic capacity of a random channel
2014, Sutter, Tobias, Sutter, David, Lygeros, John
We consider discrete memoryless channels with input and output alphabet size n whose channel transition matrix consists of entries that are independent and identically distributed according to some probability distribution v on (R≥0, B(R≥0)) before being normalized, where v is such that E[X log X) 2 1 <; ∞, μ 1 := E[X] and μ 2 := E[X log X] for a random variable X with distribution v. We prove that in the limit as n → ∞, the capacity of such a channel converges to μ 2 /μ 1 - log μ 1 almost surely and in L 2 . We further show that the capacity of these random channels converges to this asymptotic value exponentially in n.
Efficient Approximation of Quantum Channel Capacities
2016, Sutter, David, Sutter, Tobias, Mohajerin Esfahani, Peyman, Renner, Renato
We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite-dimensional output under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an additive ε-close estimate to the capacity, the presented algorithm requires O((N ν M)M3log(N)1/2 ε -1 ) steps, where N denotes the input alphabet size and M denotes the output dimension. We then generalize the method to the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite-dimensional output. This, using the idea of a universal encoder, allows us to approximate the Holevo capacity for channels with a finite-dimensional quantum mechanical input and output. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multi-dimensional integration problem. For certain families of quantum channels, we prove that the complexity to derive an additive ε-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.
Capacity approximation of memoryless channels with countable output alphabets
2014, Sutter, Tobias, Mohajerin Esfahani, Peyman, Sutter, David, Lygeros, John
We present a new algorithm, based on duality of convex programming and the specific structure of the channel capacity problem, to iteratively construct upper and lower bounds for the capacity of memoryless channels having continuous input and countable output alphabets. Under a mild assumption on the decay rate of the channel's tail, explicit bounds for the approximation error are provided. We demonstrate the applicability of our result on the discrete-time Poisson channel having a peak-power input constraint.
Efficient approximation of discrete memoryless channel capacities
2014, Sutter, David, Mohajerin Esfahani, Peyman, Sutter, Tobias, Lygeros, John
We propose an iterative method for efficiently approximating the capacity of discrete memoryless channels, possibly having additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To find an ε-approximation of the capacity, in case of no additional input constraints, the presented method has a computational complexity O(1 over εM 2 N√(logN)), where N and M denote the input and output alphabet size, and a single iteration has a complexity O(MN).